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2
index.md
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@ -26,7 +26,7 @@ material in a different way; see the [Preface](Preface).
- [Equality: Equality and equational reasoning](Equality)
- [Isomorphism: Isomorphism and embedding](Isomorphism)
- [Connectives: Conjunction, Disjunction, and Implication](Connectives)
- [Negation: Negation, with Classical and Intuitionistic Logic](Negation)
- [Negation: Negation, with Intuitionistic and Classical Logic](Negation)
- [Quantiers: Universals and Existentials](Quantifiers)
- [Lists: Lists and other data types](Lists)
- [Decidable: Booleans and decision procedures](Decidable)

28
src/Connectives.lagda
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@ -534,7 +534,6 @@ Right identity follows from commutativity of sum and left identity.
≃-∎
\end{code}
## Implication is function
Given two propositions `A` and `B`, the implication `A → B` holds if
@ -543,9 +542,9 @@ the function type, which has appeared throughout this book.
Evidence that `A → B` holds is of the form
λ{ x → N }
λ (x : A) → N x
where `N` is a term of type `B` containing as a free variable `x` of type `A`.
where `N x` is a term of type `B` containing as a free variable `x` of type `A`.
Given a term `L` providing evidence that `A → B` holds, and a term `M`
providing evidence that `A` holds, the term `L M` provides evidence that
`B` holds. In other words, evidence that `A → B` holds is a function that
@ -555,7 +554,7 @@ Put another way, if we know that `A → B` and `A` both hold,
then we may conclude that `B` holds.
\begin{code}
→-elim : ∀ {A B : Set} → (A → B) → A → B
→-elim f x = f x
→-elim L M = L M
\end{code}
In medieval times, this rule was known by the name *modus ponens*.
It corresponds to function application.
@ -566,10 +565,10 @@ while applying a function is referred to as *eliminating* the function.
Elimination followed by introduction is the identity.
\begin{code}
η-→ : ∀ {A B : Set} (f : A → B) → (λ{x → f x}) ≡ f
η-→ : ∀ {A B : Set} (f : A → B) → (λ (x : A) → f x) ≡ f
η-→ {A} {B} f = extensionality η-helper
where
η-helper : (x : A) → (λ (x′ : A) → f x′) x ≡ f x
η-helper : (x : A) → (λ (x : A) → f x) x ≡ f x
η-helper x = refl
\end{code}
The proof depends on extensionality.
@ -672,8 +671,8 @@ That is, the assertion that if either `A` holds or `B` holds then `C` holds
is the same as the assertion that if `A` holds then `C` holds and if
`B` holds then `C` holds. The proof of the left inverse requires extensionality.
\begin{code}
→-distributes-⊎ : ∀ {A B C : Set} → (A ⊎ B → C) ≃ ((A → C) × (B → C))
→-distributes-⊎ =
→-distrib-⊎ : ∀ {A B C : Set} → (A ⊎ B → C) ≃ ((A → C) × (B → C))
→-distrib-⊎ =
record
{ to = λ{ f → ( (λ{ x → f (inj₁ x) }) , (λ{ y → f (inj₂ y) }) ) }
; from = λ{ (g , h) → λ{ (inj₁ x) → g x ; (inj₂ y) → h y } }
@ -695,8 +694,8 @@ is the same as the assertion that if `A` holds then `B` holds and if
`A` holds then `C` holds. The proof of left inverse requires both extensionality
and the rule `η-×` for products.
\begin{code}
→-distributes-× : ∀ {A B C : Set} → (A → B × C) ≃ ((A → B) × (A → C))
→-distributes-× =
→-distrib-× : ∀ {A B C : Set} → (A → B × C) ≃ ((A → B) × (A → C))
→-distrib-× =
record
{ to = λ{ f → ( (λ{ x → proj₁ (f x) }) , (λ{ y → proj₂ (f y)}) ) }
; from = λ{ (g , h) → λ{ x → (g x , h x) } }
@ -711,8 +710,8 @@ and the rule `η-×` for products.
Products distributes over sum, up to isomorphism. The code to validate
this fact is similar in structure to our previous results.
\begin{code}
×-distributes-⊎ : ∀ {A B C : Set} → ((A ⊎ B) × C) ≃ ((A × C) ⊎ (B × C))
×-distributes-⊎ =
×-distrib-⊎ : ∀ {A B C : Set} → ((A ⊎ B) × C) ≃ ((A × C) ⊎ (B × C))
×-distrib-⊎ =
record
{ to = λ { ((inj₁ x) , z) → (inj₁ (x , z))
; ((inj₂ y) , z) → (inj₂ (y , z))
@ -731,8 +730,8 @@ this fact is similar in structure to our previous results.
Sums do not distribute over products up to isomorphism, but it is an embedding.
\begin{code}
⊎-distributes-× : ∀ {A B C : Set} → ((A × B) ⊎ C) ≲ ((A ⊎ C) × (B ⊎ C))
⊎-distributes-× =
⊎-distrib-× : ∀ {A B C : Set} → ((A × B) ⊎ C) ≲ ((A ⊎ C) × (B ⊎ C))
⊎-distrib-× =
record
{ to = λ { (inj₁ (x , y)) → (inj₁ x , inj₁ y)
; (inj₂ z) → (inj₂ z , inj₂ z)
@ -800,6 +799,7 @@ This chapter uses the following unicode.
⊎ U+228E MULTISET UNION (\u+)
⊤ U+22A4 DOWN TACK (\top)
⊥ U+22A5 UP TACK (\bot)
η U+03B7 GREEK SMALL LETTER ETA (\eta)
₁ U+2081 SUBSCRIPT ONE (\_1)
₂ U+2082 SUBSCRIPT TWO (\_2)
⇔ U+21D4 LEFT RIGHT DOUBLE ARROW (\<=>)

10
src/Equality.lagda
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@ -59,8 +59,9 @@ but on the left-hand side of the equation the argument has been instantiated to
which requires that `x` and `y` are the same. Hence, for the right-hand side of the equation
we need a term of type `x ≡ x`, and `refl` will do.
It is instructive to develop `sym` interactively.
To start, we supply a variable for the argument on the left, and a hole for the body on the right:
It is instructive to develop `sym` interactively. To start, we supply
a variable for the argument on the left, and a hole for the body on
the right:
sym : ∀ {ℓ} {A : Set ℓ} {x y : A} → x ≡ y → y ≡ x
sym e = {! !}
@ -75,8 +76,9 @@ If we go into the hole and type `C-C C-,` then Agda reports:
.A : Set .ℓ
.ℓ : .Agda.Primitive.Level
If in the hole we type `C-C C-C e` then Agda will instantiate `e` to all possible constructors,
with one equation for each. There is only one possible constructor:
If in the hole we type `C-C C-C e` then Agda will instantiate `e` to
all possible constructors, with one equation for each. There is only
one possible constructor:
sym : ∀ {ℓ} {A : Set ℓ} {x y : A} → x ≡ y → y ≡ x
sym refl = {! !}

106
src/Negation.lagda
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@ -1,19 +1,16 @@
---
title : "Negation: Negation, with Classical and Intuitionistic Logic"
title : "Negation: Negation, with Intuitionistic and Classical Logic"
layout : page
permalink : /Negation
---
This chapter introduces negation, and discusses intuitionistic
and classical logic.
## Imports
\begin{code}
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; sym; trans; cong)
open Eq.≡-Reasoning
open import Isomorphism using (_≃_; ≃-sym; ≃-trans; _≲_)
open Isomorphism.≃-Reasoning
open import Data.Nat using (ℕ; zero; suc; _+_; _*_)
open import Data.Nat.Properties.Simple using (+-suc)
open import Data.Empty using (⊥; ⊥-elim)
open import Data.Sum using (_⊎_; inj₁; inj₂)
open import Data.Product using (_×_; _,_; proj₁; proj₂)
@ -84,11 +81,11 @@ can conclude `¬ ¬ A` (evidenced by `¬¬-intro x`). Then from
`¬ ¬ ¬ A` and `¬ ¬ A` we have a contradiction (evidenced by
`¬¬¬x (¬¬-intro x)`. Hence we have shown `¬ A`.
Another law of logic is the *contrapositive*,
Another law of logic is *contraposition*,
stating that if `A` implies `B`, then `¬ B` implies `¬ A`.
\begin{code}
contrapositive : ∀ {A B : Set} → (A → B) → (¬ B → ¬ A)
contrapositive f ¬y x = ¬y (f x)
contraposition : ∀ {A B : Set} → (A → B) → (¬ B → ¬ A)
contraposition f ¬y x = ¬y (f x)
\end{code}
Let `f` be evidence of `A → B` and let `¬y` be evidence of `¬ B`. We
will show that assuming `A` leads to a contradiction, and hence
@ -268,7 +265,7 @@ Philip Wadler, *International Conference on Functional Programming*, 2003.)
### Exercise
Prove the following four formulas are equivalent to each other,
Prove the following three formulas are equivalent to each other,
and to the formulas `EM` and `⊎-Dual-+` given earlier.
\begin{code}
¬¬-Elim Peirce Implication : Set₁
@ -277,61 +274,34 @@ Peirce = ∀ {A B : Set} → (((A → B) → A) → A)
Implication = ∀ {A B : Set} → (A → B) → ¬ A ⊎ B
\end{code}
<!--
It turns out that an assertion such as `Set : Set` is paradoxical.
Therefore, there are an infinite number of different levels of sets,
where `Set lzero : Set lone` and `Set lone : Set ltwo`, and so on,
where `lone` is `lsuc lzero`, and `ltwo` is `lsuc lone`, analogous to
the way we represent the natural nuambers. So far, we have only used
the type `Set`, which is equivalent to `Set lzero`. Here, since each
of `double-negation` and the others defines a type, we need to use
`Set₁` as the "type of types".
-->
[NOTES]
Two halves of de Morgan's laws hold intuitionistically. The other two
halves are each equivalent to the law of double negation.
\begin{code}
dem1 : ∀ {A B : Set} → A × B → ¬ (¬ A ⊎ ¬ B)
dem1 (a , b) (inj₁ ¬a) = ¬a a
dem1 (a , b) (inj₂ ¬b) = ¬b b
dem2 : ∀ {A B : Set} → A ⊎ B → ¬ (¬ A × ¬ B)
dem2 (inj₁ a) (¬a , ¬b) = ¬a a
dem2 (inj₂ b) (¬a , ¬b) = ¬b b
\end{code}
For the other variant of De Morgan's law, one way is an isomorphism.
\begin{code}
-- dem-≃ : ∀ {A B : Set} → (¬ (A ⊎ B)) ≃ (¬ A × ¬ B)
-- dem-≃ = →-distributes-⊎
\end{code}
The other holds in only one direction.
\begin{code}
dem-half : ∀ {A B : Set} → ¬ A ⊎ ¬ B → ¬ (A × B)
dem-half (inj₁ ¬a) (a , b) = ¬a a
dem-half (inj₂ ¬b) (a , b) = ¬b b
\end{code}
The other variant does not appear to be equivalent to classical logic.
So that undermines my idea that basic propositions are either true
intuitionistically or equivalent to classical logic.
For several of the laws equivalent to classical logic, the reverse
direction holds in intuitionistic long.
\begin{code}
implication-inv : ∀ {A B : Set} → (¬ A ⊎ B) → A → B
implication-inv (inj₁ ¬a) a = ⊥-elim (¬a a)
implication-inv (inj₂ b) a = b
demorgan-inv : ∀ {A B : Set} → A ⊎ B → ¬ (¬ A × ¬ B)
demorgan-inv (inj₁ a) (¬a , ¬b) = ¬a a
demorgan-inv (inj₂ b) (¬a , ¬b) = ¬b b
\end{code}
### Exercise (`¬-stable`, `×-stable`)
Say that a formula is *stable* if double negation elimination holds for it.
\begin{code}
Stable : Set → Set
Stable A = ¬ ¬ A → A
\end{code}
Show that any negated formula is stable, and that the conjunction
of two stable formulas is stable.
\begin{code}
¬-Stable : Set₁
¬-Stable = ∀ {A : Set} → Stable (¬ A)
×-Stable : Set₁
×-Stable = ∀ {A B : Set} → Stable A → Stable B → Stable (A × B)
\end{code}
## Standard Prelude
Definitions similar to those in this chapter can be found in the standard library.
\begin{code}
import Relation.Nullary using (¬_)
import Relation.Nullary.Negation using (contraposition)
\end{code}
## Unicode
This chapter uses the following unicode.
¬ U+00AC NOT SIGN (\neg)

113
src/Quantifiers.lagda
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@ -4,6 +4,8 @@ layout : page
permalink : /Quantifiers
---
This chapter introduces universal and existential quatification.
## Imports
\begin{code}
@ -16,6 +18,8 @@ open import Data.Nat using (ℕ; zero; suc; _+_; _*_)
open import Data.Nat.Properties.Simple using (+-suc)
open import Relation.Nullary using (¬_)
open import Function using (_∘_)
open import Data.Product using (_×_; _,_; proj₁; proj₂)
open import Data.Sum using (_⊎_; inj₁; inj₂)
\end{code}
In what follows, we will occasionally require [extensionality][extensionality].
@ -29,52 +33,73 @@ postulate
## Universals
Given a variable `x` of type `A` and a proposition `B[x]` which
Given a variable `x` of type `A` and a proposition `B x` which
contains `x` as a free variable, the universally quantified
proposition `∀ (x : A) → B[x]` holds if for every term `M` of type
`A` the proposition `B[M]` holds. Here `B[M]` stands for
the proposition `B[x]` with each free occurrence of `x` replaced by
`M`. The variable `x` appears free in `B[x]` but bound in
`∀ (x : A) → B[x]`. We formalise universal quantification using the
proposition `∀ (x : A) → B x` holds if for every term `M` of type
`A` the proposition `B M` holds. Here `B M` stands for
the proposition `B x` with each free occurrence of `x` replaced by
`M`. The variable `x` appears free in `B x` but bound in
`∀ (x : A) → B x`. We formalise universal quantification using the
dependent function type, which has appeared throughout this book.
Evidence that `∀ (x : A) → B[x]` holds is of the form
Evidence that `∀ (x : A) → B x` holds is of the form
λ (x : A) → N[x]
λ (x : A) → N x
where `N[x]` is a term of type `B[x]`; here `N[x]` is a term and `B[x]`
is a proposition (or type) both containing as a free variable `x` of
type `A`. Given a term `L` providing evidence that `∀ (x : A) → B[x]`
holds and a term `M` of type `A`, the term `L M` provides evidence
that `B[M]` holds. In other words, evidence that `∀ (x : A) → B[x]`
holds is a function that converts a term `M` of type `A` into evidence
that `B[M]` holds.
where `N x` is a term of type `B x`, and `N x` and `B x` both contain
a free variable `x` of type `A`. Given a term `L` providing evidence
that `∀ (x : A) → B x` holds, and a term `M` of type `A`, the term `L
M` provides evidence that `B M` holds. In other words, evidence that
`∀ (x : A) → B x` holds is a function that converts a term `M` of type
`A` into evidence that `B M` holds.
Put another way, if we know that `∀ (x : A) → B[x]` holds and that `M`
is a term of type `A` then we may conclude that `B[M]` holds. In
medieval times, this rule was known by the name *dictum de omni*.
Put another way, if we know that `∀ (x : A) → B x` holds and that `M`
is a term of type `A` then we may conclude that `B M` holds.
\begin{code}
∀-elim : ∀ {A : Set} {B : A → Set} → (∀ (x : A) → B x) → (M : A) → B M
∀-elim L M = L M
\end{code}
In medieval times, this rule was known by the name *dictum de omni*.
If we introduce a universal and the immediately eliminate it, we can
always simplify the resulting term. Thus
(λ (x : A) → N[x]) M
simplifies to `N[M]` of type `B[M]`, where `N[M]` stands for the term
`N[x]` with each free occurrence of `x` replaced by `M` of type `A`.
Ordinary function types arise as the special case of dependent
function types where the range does not depend on a variable drawn
from the domain. When an ordinary function is viewed as evidence of
implication, both its domain and range are viewed as types of
evidence, whereas when a dependent function is viewed as evidence of a
universal, its domain is viewed as a data type and its range is viewed
as a type of evidence. This is just a matter of interpretation, since
in Agda data types and types of evidence are indistinguishable.
Unlike with conjunction, disjunction, and implication, there is no simple
analogy between universals and arithmetic.
### Exercise (`∀-distrib-×`)
Show that universals distribute over conjunction.
\begin{code}
∀-Distrib-× = ∀ {A : Set} {B C : A → Set} →
(∀ (x : A) → B x × C x) ≃ (∀ (x : A) → B x) × (∀ (x : A) → C x)
\end{code}
Compare this with the result (`→-distrib-×`) in Chapter [Connectives](Connectives).
### Exercise (`⊎∀-implies-∀⊎`)
Show that a disjunction of universals implies a universal of disjunctions.
\begin{code}
⊎∀-Implies-∀⊎ = ∀ {A : Set} { B C : A → Set } →
(∀ (x : A) → B x) ⊎ (∀ (x : A) → C x) → ∀ (x : A) → B x ⊎ C x
\end{code}
Does the converse also hold? If so, prove; if not, explain why.
## Existentials
Given a variable `x` of type `A` and a proposition `B[x]` which
Given a variable `x` of type `A` and a proposition `B x` which
contains `x` as a free variable, the existentially quantified
proposition `∃ (λ (x : A) → B[x])` holds if for some term `M` of type
`A` the proposition `B[M]` holds. Here `B[M]` stands for
the proposition `B[x]` with each free occurrence of `x` replaced by
`M`. The variable `x` appears free in `B[x]` but bound in
`∃ (λ (x : A) → B[x])`.
proposition `∃[ x ] → B x` holds if for some term `M` of type
`A` the proposition `B M` holds. Here `B M` stands for
the proposition `B x` with each free occurrence of `x` replaced by
`M`. The variable `x` appears free in `B x` but bound in
`∃[ x ] → B x`.
It is common to adopt a notation such as `∃[ x : A ]→ B[x]`,
which introduces `x` as a bound variable of type `A` that appears
@ -85,18 +110,23 @@ to introduce `x` as a bound variable.
We formalise existential quantification by declaring a suitable
inductive type.
\begin{code}
data ∃ {A : Set} (B : A → Set) : Set where
_,_ : (x : A) → B x → ∃ B
data ∃ (A : Set) (B : A → Set) : Set where
_,_ : (x : A) → B x → ∃ A B
\end{code}
Evidence that `∃ (λ (x : A) → B[x])` holds is of the form
`(M , N)` where `M` is a term of type `A`, and `N` is evidence
that `B[M]` holds.
\begin{code}
syntax ∃ A (λ x → B) = ∃[ x ∈ A ] B
\end{code}
Given evidence that `∃ (λ (x : A) → B[x])` holds, and
given evidence that `∀ (x : A) → B[x] → C` holds, where `C` does
not contain `x` as a free variable, we may conclude that `C` holds.
\begin{code}
∃-elim : ∀ {A : Set} {B : A → Set} {C : Set} → ∃ B → (∀ (x : A) → B x → C) → C
∃-elim : ∀ {A : Set} {B : A → Set} {C : Set} →
(∃[ x ∈ A ] B x) → (∀ (x : A) → B x → C) → C
∃-elim (M , N) P = P M N
\end{code}
In other words, if we know for every `x` of type `A` that `B[x]`
@ -148,7 +178,7 @@ allows us to simplify `m + suc n` to `suc (m + n)`.
Here is the proof in the forward direction.
\begin{code}
ev-ex : ∀ {n : ℕ} → even n → ∃(λ (m : ℕ) → 2 * m ≡ n)
ev-ex : ∀ {n : ℕ} → even n → ∃[ m ∈ ℕ ] (2 * m ≡ n)
ev-ex ev0 = (zero , refl)
ev-ex (ev+2 ev) with ev-ex ev
... | (m , refl) = (suc m , lemma m)
@ -171,7 +201,7 @@ return a pair consisting of `suc m` and a proof that `2 * suc m ≡ suc
Here is the proof in the reverse direction.
\begin{code}
ex-ev : ∀ {n : ℕ} → ∃(λ (m : ℕ) → 2 * m ≡ n) → even n
ex-ev : ∀ {n : ℕ} → ∃[ m ∈ ℕ ] (2 * m ≡ n) → even n
ex-ev (zero , refl) = ev0
ex-ev (suc m , refl) rewrite lemma m = ev+2 (ex-ev (m , refl))
\end{code}
@ -196,11 +226,11 @@ disjuntion and universals are generalised conjunction, this
result is analogous to the one which tells us that negation
of a disjuntion is isomorphic to a conjunction of negations.
\begin{code}
¬∃∀ : ∀ {A : Set} {B : A → Set} → (¬ ∃ (λ (x : A) → B x)) ≃ ∀ (x : A) → ¬ B x
¬∃∀ : ∀ {A : Set} {B : A → Set} → (¬ ∃[ x ∈ A ] B x) ≃ ∀ (x : A) → ¬ B x
¬∃∀ =
record
{ to = λ { ¬∃bx x bx → ¬∃bx (x , bx) }
; from = λ { ∀¬bx (x , bx) → ∀¬bx x bx }
{ to = λ { ¬∃bx x bx → ¬∃bx (x , bx) }
; from = λ { ∀¬bx (x , bx) → ∀¬bx x bx }
; from∘to = λ { ¬∃bx → extensionality (λ { (x , bx) → refl }) }
; to∘from = λ { ∀¬bx → refl }
}
@ -223,7 +253,12 @@ requires extensionality.
*Exercise* Show `∃ (λ (x : A) → ¬ B x) → ¬ (∀ (x : A) → B x)`.
## Standard Prelude
Definitions similar to those in this chapter can be found in the standard library.
\begin{code}
import Data.Product using (∃)
\end{code}
## Unicode